로그 부등식 순서쌍 개수

$| \log _{2} a - \log _{2} 10 | + \log _{2} b \leq 1$ $\displaystyle \left| \log _{2} {\frac{a}{10}} \right| \leq 1 - \log _{2} b$$\displaystyle = \log _{2} {\frac{2}{b}}$ $\displaystyle - \log _{2} {\frac{2}{b}} \leq \log _{2} \frac{a}{10} \leq \log _{2} \frac{2}{b}$ $\displaystyle \log _{2} {\frac{b}{2}} \leq \log _{2} \frac{a}{10} \leq \log _{2} \frac{2}{b}$ $\displaystyle \rightarrow \frac{b}{2} \leq \frac{a}{10} \leq \frac{2}{b}$ $\therefore 5 b \leq a , ab \leq 20$ $5 b ^{2} \leq ab ,$ 즉, $5 b ^{2} \leq 20$ $\rightarrow b ^{2} \leq 4$ $b$는 자연수이므로 $b = 1$ 또는 $b = 2$ $b = 1$이면 $5 \leq a \leq 20$ $\therefore a$는 $16$개 $b = 2$이면 $10 \leq a \leq 10$ $\therefore a$는 $1$개 따라서 순서쌍 $( a , b )$는 $17$개